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Basic Forms and Orbit Spaces: a Diffeological Approach Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 026, 19 pages Basic Forms and Orbit Spaces: a Diffeological Approach Yael KARSHON y and Jordan WATTS z y Department of Mathematics, University of Toronto, 40 St. George Street, Toronto Ontario M5S 2E4, Canada E-mail: [email protected] URL: http://www.math.toronto.edu/karshon/ z Department of Mathematics, University of Colorado Boulder, Campus Box 395, Boulder, CO, 80309, USA E-mail: [email protected] URL: http://euclid.colorado.edu/~jowa8403 Received October 06, 2015, in final form February 16, 2016; Published online March 08, 2016 http://dx.doi.org/10.3842/SIGMA.2016.026 Abstract. If a Lie group acts on a manifold freely and properly, pulling back by the quotient map gives an isomorphism between the differential forms on the quotient manifold and the basic differential forms upstairs. We show that this result remains true for actions that are not necessarily free nor proper, as long as the identity component acts properly, where on the quotient space we take differential forms in the diffeological sense. Key words: diffeology; Lie group actions; orbit space; basic differential forms 2010 Mathematics Subject Classification: 58D19; 57R99 1 Introduction Let M be a smooth manifold and G a Lie group acting on M.A basic differential form on M is a differential form that is G-invariant and horizontal; the latter means that evaluating the form on any vector that is tangent to a G-orbit yields 0. Basic differential forms constitute a subcomplex of the de Rham complex. If G acts properly and with a constant orbit-type, then the quotient M=G is a manifold, and, denoting the quotient map by π : M ! M=G, the pullback by this map gives an isomorphism of the de Rham complex on M=G with the complex of basic forms on M. Even if M=G is not a manifold, if G acts properly, then the cohomology of the complex of basic forms is isomorphic to the singular cohomology of M=G with real coefficients; this was shown by Koszul in 1953 [17] for compact group actions and by Palais in 1961 [19] for proper group actions. In light of these facts, some authors define the de Rham complex on M=G to be the complex of basic forms on M. There is another, intrinsic, definition of a differential form on M=G, which comes from viewing M=G as a diffeological space (see Section2). This definition agrees with the usual one when M=G is a manifold. Differential forms on diffeological spaces admit exterior derivatives, wedge products, and pullbacks under smooth maps. Spaces of differential forms are themselves diffeological spaces too. With this notion, here is our main result: Theorem 1.1. Let G be a Lie group acting on a manifold M. Let π : M ! M=G be the quotient map. (i) The pullback map π∗ :Ω∗(M=G) ! Ω∗(M) is one-to-one. Its image is contained in the ∗ ∗ space Ωbasic(M) of basic forms. As a map to its image, the map π is an isomorphism of differential graded algebras and a diffeological diffeomorphism. 2 Y. Karshon and J. Watts (ii) If the restriction of the action to the identity component of G is proper, then the image of the pullback map is equal to the space of basic forms: ∼ ∗ ∗ = ∗ π :Ω (M=G) / Ωbasic(M) : We prove Theorem 1.1 in Section5, after the proof of Proposition 5.10. Remark 1.2. 1. Part (i) of Theorem 1.1, which follows from the results of Sections2 and3, is not difficult. The technical heart of Theorem 1.1 is Part (ii), which is proved in Proposition 5.10: if the identity component of the group acts properly, then every basic form on M descends to a diffeological form on M=G. This fact is non-trivial even when the group G is finite. 2. The quotient M=G can be non-Hausdorff. Nevertheless, even if its topology is trivial, M=G may have non-trivial differential forms, and its de Rham cohomology may be non-trivial. See, for example, the irrational torus in Remark 5.12. 3. We expect the conclusion of Part (ii) of Theorem 1.1 to hold under more general hypothe- ses. In particular, our assumption that the identity component of G act properly on M is sufficient but not necessary; see, for instance, Example 5.13. Diffeology was developed by Jean-Marie Souriau (see [26]) around 1980, following earlier work of Kuo-Tsai Chen (see, e.g., [4,5]). Our primary reference for this theory is the book [14] by Iglesias-Zemmour. In many applications, diffeology can serve as a replacement for the manifold structures (modelled on locally convex topological vector spaces) on spaces of smooth paths, functions, or differential forms. See, for example, [2]. The category of diffeological spaces is complete and co-complete (see [1]); in particular, subsets and quotients naturally inherit diffeological structures. It is also Cartesian closed, where we equip spaces of smooth maps with their natural functional diffeology. It is also common to consider M=G as a (Sikorski) differential space, by equipping it with the set of those real valued functions whose pullback to M is smooth. See Hochschild [12], Bredon [3], G. Schwarz [22], and Cushman and Sniatycki´ [7]. This structure is determined by n the diffeology on M=G but is weaker. For example, the quotients R =SO(n) for different positive integers n are isomorphic as differential spaces but not as diffeological spaces. (See Exercise 50 of Iglesias [14] with solution at the end of the book.) There are several inequivalent notions of \differential form" on differential spaces (see Sniatycki´ [25] and Watts [27]); we do not know of an analogue of Theorem 1.1 for any of these notions. Turning to higher category theory, we can also consider the stack quotient [M=G]. It is a differentiable stack over the site of manifolds, and it is represented by the action groupoid G × M⇒M. One can define a differential k-form on the stack [M=G] as a map of stacks from [M=G] to the stack of differential forms Ωk. In [30], Watts and Wolbert define a functor Coarse from stacks to diffeological spaces for which Coarse([M=G]) is equal to M=G equipped with the quotient diffeology. In this language, Theorem 1.1 gives an isomorphism Ωk([M=G]) =∼ Ωk(Coarse([M=G])); when the identity component of G acts properly. We note that a diffeological space can also be viewed as a stack, and applying Coarse recovers the original diffeological space. We also note that the quotient stack [M=G] often contains more information than the quotient diffeology; for example, if M is a point, the quotient diffeological space is a point, but the stack [M=G] determines the group G. Finally, we note that Karshon and Zoghi [16] give sufficient conditions Basic Forms and Orbit Spaces: a Diffeological Approach 3 for a Lie groupoid to be determined up to Morita equivalence by its underlying diffeological space. In the special case that the Lie group G is compact, the main results of this paper appeared in the Ph.D. Thesis of the second author [28], supervised by the first author. A generalisation to proper Lie groupoids, which relies on these results and on a deep theorem of Crainic and Struchiner [6] showing that all proper Lie groupoids are linearisable, was worked out by Watts in [29]. The paper is structured as follows. For the convenience of the reader, Section2 contains background on diffeology, differential forms on diffeological spaces, and Lie group actions in connection to diffeology. Section3 contains a proof that the pullback map from the space of differential forms on the orbit space M=G to the space of differential forms on the manifold M is an injection into the space of basic forms, and is a diffeomorphism onto its image. Section4 contains a technical lemma: \quotient in stages". Section5 is the technical heart of the paper; it contains a proof that, if the identity component of the group acts properly, then the pullback map surjects onto the space of basic forms. Our appendices contain two applications of the special case of Theorem 1.1 when the group G is finite. In AppendixA we show that, on an orbifold, the notion of a diffeological differential form agrees with the usual notion of a differential form on the orbifold. In AppendixB we show that, on a regular symplectic quotient (which is also an orbifold), the notion of a diffeological differential form also agrees with Sjamaar's notion of a differential form on the symplectic quotient. The case of non-regular symplectic quotients is open. 2 Background on diffeological spaces In this section we review the basics of diffeology, diffeological differential forms, and Lie group actions in the context of diffeology. For more details (e.g., the quotient diffeology is in fact a diffeology), see Iglesias-Zemmour [14]. The basics of diffeology This subsection contains a review of the basics of diffeology; in particular, the definition of a diffeology and diffeologically smooth maps, as well as various constructions in the diffeological category. Definition 2.1 (diffeology). Let X be a set. A parametrisation on X is a function p: U ! X n where U is an open subset of R for some n.A diffeology D on X is a set of parametrisations that satisfies the following three conditions. 1.( Covering) For every point x 2 X and every non-negative integer n 2 N, the constant n function p: R ! fxg ⊆ X is in D.
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